Problem 7 - Advanced Engineering Mathematics

Question

\( \mathbf{r}(t)\) is the position vector of a moving particle. Graph the curve and the velocity and acceleration vectors at the indicated time. Find the speed at that time. $$ \mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k} ; t=1 $$

Step-by-Step Solution

Answer

At time \( t=1 \), the speed is \( \sqrt{14} \).

Step 1: Identify the Position Vector

The position vector of the particle is given by \( \mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k} \). At time \( t = 1 \), the position vector becomes \( \mathbf{r}(1) = 1 \mathbf{i} + 1^2 \mathbf{j} + 1^3 \mathbf{k} = \mathbf{i} + \mathbf{j} + \mathbf{k} \).

Step 2: Calculate the Velocity Vector

The velocity vector \( \mathbf{v}(t) \) is the derivative of the position vector \( \mathbf{r}(t) \). Differentiate each component: \( \mathbf{v}(t) = \frac{d}{dt}(t) \mathbf{i} + \frac{d}{dt}(t^2) \mathbf{j} + \frac{d}{dt}(t^3) \mathbf{k} = 1 \mathbf{i} + 2t \mathbf{j} + 3t^2\mathbf{k} \). At \( t = 1 \), \( \mathbf{v}(1) = 1 \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k} \).

Step 3: Calculate the Acceleration Vector

The acceleration vector \( \mathbf{a}(t) \) is the derivative of the velocity vector \( \mathbf{v}(t) \). Differentiate each component: \( \mathbf{a}(t) = \frac{d}{dt}(1) \mathbf{i} + \frac{d}{dt}(2t) \mathbf{j} + \frac{d}{dt}(3t^2) \mathbf{k} = 0\mathbf{i} + 2\mathbf{j} + 6t\mathbf{k} \). At \( t = 1 \), \( \mathbf{a}(1) = 0\mathbf{i} + 2\mathbf{j} + 6\mathbf{k} \).

Step 4: Graph the Curve and Vectors

On a 3D graph, plot the position vector endpoint (1, 1, 1), the velocity vector (1, 2, 3), and the acceleration vector (0, 2, 6) at the point (1, 1, 1). The velocity and acceleration vectors originate from this point to represent the change.

Step 5: Calculate the Speed

The speed of the particle is the magnitude of the velocity vector. Calculate it using: \( \text{Speed} = \| \mathbf{v}(1) \| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \).

Key Concepts

Velocity vector

Understanding vectors is crucial when analyzing motion. The velocity vector represents a particle's direction and rate of change in its position over time. To find the velocity vector, you take the derivative of the position vector with respect to time.

In this exercise, the position vector is given by: \[ \mathbf{r}(t) = t \mathbf{i} + t^2 \mathbf{j} + t^3 \mathbf{k} \]

• First, differentiate each component of the position vector: \( \frac{d}{dt}(t) = 1 \), \( \frac{d}{dt}(t^2) = 2t \), and \( \frac{d}{dt}(t^3) = 3t^2 \).
• This results in the velocity vector: \[ \mathbf{v}(t) = 1 \mathbf{i} + 2t \mathbf{j} + 3t^2 \mathbf{k} \]

At a specific time, say \( t = 1 \), we plug in the value to find the velocity. So, \( \mathbf{v}(1) = 1 \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k} \).

Acceleration vector

The acceleration vector is a measure of how quickly the velocity of a particle changes with time. It is the derivative of the velocity vector.

Given the velocity vector\[\mathbf{v}(t) = 1 \mathbf{i} + 2t \mathbf{j} + 3t^2 \mathbf{k} \]

• Differentiate each component to find the acceleration: \( \frac{d}{dt}(1) = 0 \), \( \frac{d}{dt}(2t) = 2 \), and \( \frac{d}{dt}(3t^2) = 6t \).
• This gives the acceleration vector: \[ \mathbf{a}(t) = 0 \mathbf{i} + 2 \mathbf{j} + 6t \mathbf{k} \]

At \( t = 1 \), substitute into the equation to find \( \mathbf{a}(1) = 0 \mathbf{i} + 2 \mathbf{j} + 6 \mathbf{k} \). This indicates the acceleration at that moment in time.

Speed calculation

Speed is a scalar quantity that represents the magnitude of the velocity vector. To calculate speed, you find the length or magnitude of the velocity vector at the given time.

Using our velocity vector at \( t = 1 \)\[ \mathbf{v}(1) = 1 \mathbf{i} + 2 \mathbf{j} + 3 \mathbf{k} \]

• We apply the formula for magnitude: \[ \text{Magnitude of } \mathbf{v}(1) = \sqrt{1^2 + 2^2 + 3^2} \]
• This simplifies to: \[ \sqrt{1 + 4 + 9} = \sqrt{14} \]

Therefore, the speed at \( t = 1 \) is \( \sqrt{14} \), which represents the rate at which the particle is moving along its path.

3D graph plotting

Graphing vectors and the path of a moving particle helps visualize the behavior and interactions of these components in space.

To create a 3D graph for this exercise:

• Plot the position vector's endpoint at \( (1, 1, 1) \), which is the particle's position at \( t = 1 \).
• From this point, draw the velocity vector \( (1, 2, 3) \). This shows the direction and speed of movement.
• Also, draw the acceleration vector \( (0, 2, 6) \) originating from the same position, indicating how the velocity is changing over time.

3D graph plotting allows you to see how these vectors interact dynamically as the particle moves through three-dimensional space, providing a clearer understanding of motion.